Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #213 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D4⋊(C3⋊C16) |
| Sylow | C3 ⋊ D4⋊C16 |
110 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(5,8)(6,11)(9,10);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27)Polycyclic presentation:
2^7,3|a^2=d,c^2=e,d^2=f,f^2=g,b^a=bc,c^a=ce,h^a=h^2,c^b=ce96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4⋊C16, C2×D4⋊(C3⋊C16), #10704888, … |
| maximal normal subgroup of | D3×D4⋊C16, C2×D4⋊(C3⋊C16), #10704888, … |
| maximal quotient of | #10681836, #10681839, #10681842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 336);